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Article: Humphries, M.D. and Gurney, K. (2008) Network 'small-world-ness': a quantitative method for determining canonical network equivalence. Plos One, 3 (4). Art No.e0002051. ISSN 1932-6203 https://doi.org/10.1371/journal.pone.0002051

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[email protected] https://eprints.whiterose.ac.uk/ Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence

Mark D. Humphries*, Kevin Gurney Adaptive Behaviour Research Group, Department of Psychology, University of Sheffield, Sheffield, United Kingdom

Abstract

Background: Many technological, biological, social, and information networks fall into the broad class of ‘small-world’ networks: they have tightly interconnected clusters of nodes, and a shortest mean length that is similar to a matched (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction (‘small/not-small’) rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network’s small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model – the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.

Methodology/Principal Findings: We defined a precise measure of ‘small-world-ness’ S based on the trade off between high local clustering and short path length. A network is now deemed a ‘small-world’ if S.1 - an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.

Conclusions/Significance: We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.

Citation: Humphries MD, Gurney K (2008) Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence. PLoS ONE 3(4): e0002051. doi:10.1371/journal.pone.0002051 Editor: Olaf Sporns, Indiana University, United States of America Received November 21, 2007; Accepted March 16, 2008; Published April 30, 2008 Copyright: ß 2008 Humphries, Gurney. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: MDH acknowledges support by the EU Framework 6 IST-027819-IP (ICEA) project. KG acknowledges the support of EPSRC grant EP/C516303/1. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: [email protected]

Introduction However, the existing ‘small-world’ definition is a categorical one, and breaks the continuum of network topologies into the Networks are widely used to both represent real-world systems three classes of regular, random, and small-world networks, with for topological study [1] and as a substrate for modeling their the latter being the broadest. It is unclear to what extent the real- dynamics [2]. Many real technological, biological, social, and world systems in the small-world class have common network information networks fall into the broad class of ‘small-world’ properties and to what specific point in the ‘‘middle-ground’’ networks [3], a middle ground between regular and random (between random and regular) a network generating model must networks: they have high local clustering of elements, like regular be tuned to genuinely capture the topology of such systems. Here networks, but also short path lengths between elements, like we explore a continuous, quantitative, measure of ‘small-world- random networks. Membership of the ‘small-world’ network class ness’, with the aim of overcoming these inadequacies in the also implies that the corresponding systems have dynamic current theory of small-world networks. properties different from those of equivalent random or regular networks [3–7]. Network formalism One popular method for studying small-world networks is to use When describing a real-world system as a network, each an equivalent network model to generate other similar instances of element of the system is represented by a or node, and the class of systems under study. Such generating models may also relationships or interactions between elements are represented by possess analytic properties that, we assume, may be extrapolated to edges between nodes. Two nodes are said to be neighbors if they the target system. One canonical model used as a candidate for are connected by an edge, and the ki of node i is the number network equivalence is the original Watts-Strogatz (WS) model, of neighbors it has. The minimum path length between two nodes is which has been used as a substrate for studying dynamics in the the minimum number of edges that must be traversed to get from diverse fields of ecology [8], economics [9,10], epidemiology one node to the other. The mean value of the minimum path [11,12], and neuroscience [13]. length over all node pairs will be denoted by L.

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A key concept in defining small-worlds networks is that of The categorical definition of small-world network above implies $ D& D. ‘clustering’ which measures the extent to which the neighbors of a lg 1 and cg 1, which, in turn, gives S 1. We can, therefore, node are also interconnected. Watts and Strogatz [3] defined the now make a quantitative categorical definition of a ‘small-world’ ws ci of node i by network Definition 2. A network is said to be a small-world network if 2E SD.1 ws~ i ci 1 ws ki ki{1 ð Þ A similar definition may also be given with respect to S . ðÞ However, notwithstanding the new categorical definition, we where Ei is the number of edges between the neighbors of i. The wish to emphasize here the utility of using a continuously graded ws ws clustering coefficient of the network C is then the mean of ci notion of small-world-ness. We go on, therefore, to analyze the over all nodes. An alternative definition of network clustering in properties of the new metrics, and apply them to real-world data common use [14], based on transitivity, is expressed by for the first time (in [16] we originally proposed this metric as a tool for comparing theoretical neuroanatomy models; its subse- 3|number of triangles quent adoption by others [17,18,19] motivated us to consider its CD~ , 2 theoretical and empirical applications as a universal metric). number of paths of length 2 ð Þ where a ‘triangle’ is a set of three nodes in which each contacts the Results other two. Both capture intuitive notions of clustering but, though often in good agreement, values for Cws and CD can differ by an New metrics behave as required with the Watts-Strogatz order of magnitude for some networks. We consider mainly CD model D here, but report where using Cws leads to different results. We first checked that the metric S behaves as required on the A network G with n nodes and m edges is a small-world network canonical Watts-Strogatz (WS) model of small-world generation [3] if it has a similar path length but greater clustering of nodes [3]. The WS model begins with a ring of n nodes, each node than an equivalent Erdo¨s-Re´nyi (E–R) random graph [15] with connected to its nearest neighbors out to some range K. Each edge the same m and n (an E–R graph is constructed by uniquely in turn is ‘re-wired’ to a new target node with probability p assigning each edge to a node pair with uniform probability). More (Figure 1A). Values of p = 0 and p = 1 give regular and random D networks, respectively, with intermediate p values resulting in formally, let Lg be the mean shortest path length of G and Cg its D ‘small-world’ networks that share properties of both provided that clustering coefficient using (2). Let Lrand and Crand be the corresponding quantities for the corresponding E–R random the network is connected and sparse — densely connected graph. These ideas may be used to supply a semi-quantitative networks trivially have small mean path lengths and high categorical definition of a small world network [3] clustering coefficients. Definition 1. The network G is said to be a small-world Figure 1 shows that small-world-ness captures the topology $ D& D changes: it has a unique maximum at intermediate values of the network if Lg Lrand and Cg Crand . Here a similar definition applies if we use (1) to define clustering re-wiring parameter p, indicating the maximum trade-off between coefficients. high clustering and low path length (Figure 1B), and decays with increasing edge density for a fixed size of network, reflecting the requirement of sparseness (Figure 1C). We can see why this occurs New measures of small-world-ness for increasing density. The edge density of a network is given by Put

2m CD j~ : 8 D~ g { cg D 3 nn 1 ð Þ Crand ð Þ ðÞ and As jR1 then both Cws, CDR1 and LR1 because all nodes become connected; and as this would apply for both a given real- Lg D lg~ 4 world network and its E-R random graph equivalent, so S , Lrand ð Þ SwsR1 regardless of n: high edge density results in low small- world-ness. We then define a quantitative metric of ‘small-world-ness’ SD according to Small-world-ness scales linearly with n for real networks We computed SD and Sws for a broad range of technological, D cg biological, social, and information networks (33 networks in total; SD~ 5 lg ð Þ Table 1, see Materials and Methods). To our surprise, we found that both forms of small-world-ness scale linearly with the size of In a similar way, putting the network across all systems falling into the small-world class (Figure 2A,B), irrespective of their originating domain or their ws ws Cg other topological properties (e.g. their , degree c ~ 6 D D g ws correlation). For S , it was not possible to find or calculate C (and Crand ð Þ hence SD) for 6 of the 33 networks. However, for the remaining 27, we define Sws all had SD.1 and were therefore deemed to be small-world in the new scheme (Definition 2). To ensure the robustness of the ws categorization, networks with borderline values 1#SD#3 were ws cg S ~ 7 tested for significance using Monte Carlo sampling of 1000 lg ð Þ

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Figure 1. Small-world-ness S behaves as required on the Watts-Strogatz (WS) [3] model of small-world networks. A The WS model begins with a ring of n nodes, each node connected to its nearest neighbors out to some range K (here K = 3). Each edge in turn is re-wired to a new target node with probability p. B The WS model shows that p = 0 gives a regular network, with high clustering but high path length; p = 1 gives a pseudo-random network, with low clustering and path length; and intermediate p values give small-world networks with high clustering and low path lengths. The S metric tracks these changes precisely, and shows which unique p value corresponds to high clustering and low path length. Normalized L (,); normalized C (#); normalized S (N). C The small-world property only applies to sparse networks: densely connected networks trivially have high clustering and short path lengths. With increasing edge density j on the WS model, the S metric indicates the absence of meaningful small-world structure. A particular edge density for the WS model is obtained by setting K =[j(n21)/2]. All numerical results obtained on graphs with n = 1000, each data-point an average over 20 realizations for each p (with K = 10) or j (with p = 0.1) value. doi:10.1371/journal.pone.0002051.g001 equivalent E–R random graphs for each network, estimating 99% relationships are held only approximately, but because these confidence intervals using standard methods (see Materials and approximations are often very good we show them as equalities. Methods). All such networks had small-world-ness scores signifi- Note that from here on we use subscript names to identify analytic cantly greater than an equivalent E–R random graph. For the 27 quantities that pertain to a particular network model only. D. networks for which S 1, linear regression on log-transformed If Lws is the mean shortest path length in the WS model, then it quantities (see Materials and Methods) allowed an estimate of the is known [14] that 2 best power law fit: SD = 0.023n0.96 (r2 = 0.78; p =3610 9). This is an essentially linear scaling of SD with n. ws n 1 {1 x For S , 3 networks in our data-set were not small-worlds: Lws~ f nKp , where fx~ tanh 9 ws K 2z xz2 relationships amongst students [20] S = 0.27 (network #9); a ðÞ ðÞ 2px 2x rffiffiffiffiffiffiffiffiffiffi ð Þ ws ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi freshwater [21,1] S = 0.74 (network #28); and the E- Similarly, it is known for E-R random graphs [23] that Coli reaction graph [22] Sws = 0.67 (network #32). This demonstrates both that the small-world property is not robustly achieved for small networks, and that it is contingent on the ln n Lrand ~ ðÞ 10 particular measure of clustering used. Once again, networks with ln SkT ð Þ borderline values 1#Sws#3 were tested using Monte Carlo ðÞ methods for significant membership of the small-world category where Ækæ is the expected value of the degree across the network. In and were found to satisfy this criterion. For the 30 networks with the WS model, node degree and range are related by Ækæ =2K,sothat Sws.1, a similar regression to that used for SD gave 2 Sws = 0.012n1.11 (r2 = 0.84; p = 1.3610 11). Thus, there is also a ws ln n robust linear scaling of S with n (see Text S1 for further details). Lrand ~ ðÞ 11 ln 2K ð Þ ðÞ

Linear scaling on the Watts-Strogatz model Using (11), (9) and (4) the path length ratio lws for the WS model is Linear scaling of small-world-ness with network size was unexpectedly shown by the canonical Watts-Strogatz model [3] of small-world network generation. We now show that this result nln 2K fnKp lws~ ðÞðÞ 12 can be explained analytically. In what follows, many of the Kln n ð Þ ðÞ

PLoS ONE | www.plosone.org 3 April 2008 | Volume 3 | Issue 4 | e0002051 Table 1. Table of small-world-ness values and other topological properties of real networks. Network ‘Small-World-Ness’

D ws D ws class # network nmÆkæ j LCC S S p (WS) Reference

Social 1 Dolphins{ 62 159 5.13 0.084 3.36 0.31 0.26 2.8 2.35 0.64 [41] 2 2 film actors 449913 25516482 113.43 2.5610 4 3.48 0.2 0.78 627 2446 0.95 [1,3] 3 company directors 7673 55392 14.44 0.002 4.6 0.59 0.88 228 341 0.77 [1,23] 2 4 math coauthorship 253339 496489 3.92 1.6610 5 7.57 0.15 0.34 11666 26443 0.7 [1,45] 2 5 coauthorship 52909 245300 9.27 1.8610 4 6.19 0.45 0.56 2026 2521 0.73 [1,46] 2 6 biology coauthorship 1520251 11803064 15.53 1610 5 4.92 0.088 0.6 9089 61967 0.88 [1,46] 2 7 email messages 59912 86300 1.44 4.8610 5 4.95 - 0.16 - 40524 n/a [1,47] 2 8 email address books 16881 57029 3.38 4610 4 5.22 0.17 0.13 1301 995 0.64 [1,48] 9 student relationships 573 477 1.67 0.0029 16.01 0.005 0.001 1.34 0.27 n/a [1,20] 10 newspaper article co- 459 1422 6.2 0.0135 2.98 - 0.02 - 1.67 n/a [49] occurence 11 US directors 11057 74414 13.46 0.0012 5.19 0.56 0.87 315 494 0.77 [50] 12 UK directors 8850 39741 8.98 0.001 6.46 0.61 0.89 386 561 0.71 [50] 13 German directors 4185 30438 14.55 0.0035 6.4 0.72 0.93 100.71 129.7 0.79 [50] 2 Information 14 WWW nd.edu 269504 1497135 5.56 4610 5 11.27 0.11 0.29 3453 9104 0.81 [1,51] 15 Roget’s Thesaurus 1022 5103 4.99 0.0098 4.87 0.13 0.15 23.54 27.17 0.76 [1,52] 16 word adjacency{ 112 425 7.59 0.0684 2.54 0.16 0.17 2.13 2.34 0.74 [26] 17 book purchases{ 105 441 8.4 0.081 3.08 0.35 0.49 3.09 4.33 0.71 V.Kreb, unpublished (www.orgnet. com) 2 Technological 18 10697 31992 5.98 5.6610 4 3.31 0.035 0.39 98.09 1093 0.83 [1,53] 2 19 power grid 4941 6594 2.67 5.4610 4 18.99 0.1 0.08 84.45 67.56 0.8 [1,3] 20 train routes 587 19603 66.79 0.114 2.16 - 0.69 - 4.26 n/a [1,54] 21 software packages 1439 1723 1.2 0.0017 2.42 0.07 0.082 1403 1644 n/a [1,25] 22 software classes 1377 2213 1.61 0.0023 1.51 0.033 0.012 285.26 103.73 n/a [1,55] 2 23 electronic circuits 24097 53248 4.42 1.8610 4 11.05 0.01 0.03 33.5 100.5 0.91 [1,56] 24 peer-to-peer network 880 1296 2.95 0.0034 4.28 0.012 0.011 5.26 4.82 0.85 [1,57] Biological 25 765 3686 9.65 0.0126 2.56 0.09 0.67 8.18 60.89 0.82 [1,58] 26 yeast protein interactions 2115 2240 0.001 2.12 6.8 0.072 0.071 107.85 106.35 0.73 [1,59] 27 marine food web 135 598 4.43 0.0661 2.05 0.16 0.23 7.84 11.27 0.64 [1,60] 28 freshwater food web 92 997 10.84 0.2382 1.9 0.2 0.087 1.7 0.74 0.74 [1,21] 29 C.Elegans{ 277 1918 13.85 0.05 2.64 0.2 0.28 3.21 4.51 0.81 [42] 30 Macaque cortex{ 95 1522 32.04 0.34 1.78 0.7 0.77 1.53 1.69 0.79 [42] 31 E. Coli substrate 282 1036 7.35 0.0261 2.9 - 0.59 - 22.08 n/a [22] 32 E. Coli reaction 315 8915 56.6 0.18 2.62 - 0.22 - 0.67 n/a [22] 33 functional cortical 90 405 9 0.1 2.49 - 0.53 - 4.32 n/a [17] connectivity

Entries ‘-’ indicate missing data; n/a indicates values that could not be computed. All SD,Sws, edge density j and implied p(WS) were computed by us; for networks marked { we have computed some or all of Ækæ, L, CD and Cws from available data-sets. References are given for the source of the original network data, and also for the analyses where these were done separately. doi:10.1371/journal.pone.0002051.t001

D ~ The function f(x) in (9) has an upper asymptote of ln(2x)/4x if Crand SkT=n, so that using Ækæ =2K nKp&1. Thus, assuming nKp&1,(12)becomes 2K CD ~ 15 ln 2nKp ln 2K rand n ð Þ lws~ ðÞðÞ 13 4K2pln n ð Þ ðÞ Therefore, using, (14), (15) and (3) If CD is the clustering coefficient of the WS model (using (2) as the ws 3 K{1 metric), then it is known [24] that cD ~ ðÞ1{p 3n 16 ws 4K 2K{1 ðÞ ð Þ ðÞ 3 K{1 From (5), (16) and (13), CD ~ ðÞ1{p 3 14 ws 22K{1 ðÞ ð Þ ðÞ ln n SD ~hK,p ðÞ n 17 For E–R random graphs [23], to a good approximation, ws ðÞln 2Kp zln n ð Þ ðÞðÞ

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Figure 2. Correlation of real-world network properties. A Small-world-ness SD scales linearly with network size n across real networks from all domains, and irrespective of their other properties. We show SD for all 27 networks for which CD could be found or calculated; result was 2 SD = 0.023n0.96 (r2 = 0.78; p =3610 9). The dashed line is the theoretical maximum small-world-ness value of SD = 0.181n (see text), given the implied mean degree of Ækæ<5 (see below). B Similarly, using known or calculated Cws, we found Sws = 0.012n1.11 (30 networks with Sws.1; r2 = 0.84; 2 p = 1.3610 11). C Number of edges m also scales linearly with network size n — CD data-set shown. Best-fit model was m = 2.46n1.06 (27 networks, 2 r2 = 0.92, p =4610 15), implying a mean node degree of Ækæ =2m/n<5. For the Cws data-set, we found m = 3.16n1.03 (30 networks, r2 = 0.91, 215 p =2610 ) implying Ækæ<6.32. Residuals of all regressions on log10-transformed data did not significantly differ from a normal distribution at p = 0.01 (Anderson-Darling test [44]. CD data-set, 27 networks: n vs SD: A2 = 0.36, p = 0.5; n vs m: A2 = 0.45, p = 0.28. Cws data-set, 30 networks: n vs Sws: A2 = 0.8, p = 0.04; n vs m: A2 = 0.82, p = 0.033). Network domains: social (,); information (); technological (#); biological (X). doi:10.1371/journal.pone.0002051.g002 where h(K, p) is a function of K and p only. The term in the square with SD.1, the best fit model was m = 2.46n1.06 (27 networks, 2 brackets tends to 1 as nR‘ and so, for large enough n, SD for the WS r2 = 0.92, p =4610 15), implying a mean node degree of Ækæ =2m/ model scales with n. To quantify this approximation, we performed a n<5; for networks with Sws.1, the best fit model was m = 3.16n1.03 2 linear regression on log-transformed quantities (just as for the real (30 networks, r2 = 0.91, p =2610 15) implying Ækæ<6.32. Thus, the networks) over the typical range of n encountered in our sample of real-world networks fulfill the prediction of constant mean node 2 networks, 102#n#107, and found a linear fit, with r2 within 10 5 of degree. A similar result holds for p values; we found that all unity. testable real-world systems fall into a very limited range of p for the equivalent WS model (0.64#p#0.95 and sp = 0.0806). Establishing the precise WS model correlate of a real An alternative view of these results is as follows. We could start network with the empirically observed approximate constancy of mean The WS network is often used as a generative model for real node degree Ækæ and calculated rewiring parameter p for the real D small-world networks [e.g. 8–13]. This is assumed to establish a ‘first- world networks, and deduce a linear scaling of S for the WS D pass’ model of that system’s topology, which may be augmented by models. Then, under the equivalence of S for both real-world considering other factors such as degree sequence [23], degree networks and their WS counterparts, we could have predicted that D correlation [25], [26] and other properties. S for the real-world networks would also scale linearly. In matching the WS parameters K, p, n to the target system, we know n, can measure Ækæ (giving K = Ækæ/2), but estimating p has, The linear scaling of small-world-ness with n is not hitherto, remained problematic. However, using our new metric of small-world-ness, it is possible to establish p in a principled way. inevitable Is the relationship S n inevitable for all systems? (The Thus, if G is a real (target) network with measured small-world- / D subsequent argument holds for S based on either definition of ness Sg , we identify it with the WS network with the same value of D ~ D { D D clustering coefficient and so superscripts D, ws are dropped). To S . That is, form eK,p,n Sws K,p,n Sg , where Sws K,p,n is given by the right handðÞ side of (17),ðÞ and minimise e withðÞ respect to investigate this we note that it is always possible to write Si = aini p, keeping K, n at their measured values. We did this for our for the ith system, for some value ai; in the case of linear scaling, ai sample of real-world systems, omitting those for which Ækæ#2 since is constant. To proceed further, we now express ai in terms of D other system parameters. Using the definition of S and (10) for the expressions used in defining Sws are inaccurate in these cases (we used Matlab routine fzero, initial value of p = 0.5). The random graphs, resulting p values for the equivalent WS model are listed in Table 1 D Given that the real-world networks showed S n, the WS CiLrand Ci nilnni / Si~ ~ 18 networks derived from them under the procedure described here LiCrand Li SkTilnSkTi ð Þ must do likewise (they have identical SD values). However, the result in the previous section would suggest that this implies K, p where Ci, Li, Ækæi are the clustering coefficient, mean shortest path are roughly constant for this set of WS networks. length, and mean node degree of system i respectively. While we To investigate the constancy of K we used the result that do not know exactly how Li depends on n, we note that the mean Ækæ =2m/n (where m is the number of edges in the network). So, shortest path length for small-world networks is usually assumed to using K = Ækæ/2, constant K is equivalent to establishing m n. scale logarithmically like random graphs: from (11), Lrand = [1/ / Figure 2C shows the result of regressing m against n (using log- ln(Ækæ)]ln(n); and for the WS model, using (9) with large n, Lws = (1/ transformed quantities) for the real world networks. For networks 4K2p)ln(n). Both relations are of the form L = bln(n) where b is

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independent of n. We therefore write Li = biln(ni), where bi is the factor that ensures the equality to be true (i.e it plays a similar role in this respect as ai). This gives

Ci ai~ 19 biSkTilnSkTi ð Þ In general, there is no a priori reason to suppose that the variables Ci, Ækæi and bi are either all constant, or co-vary in a way commensurate with constancy for ai. However, for the sample of networks used here, as noted above, the mean node degree Ækæi is approximately constant. It is now instructive to see how much co- variation is required between the remaining two variables in order to ensure a significantly different power law holds between S and n. 1.5 Figure 3. Robustness of WS model prediction S n. We test the Thus, suppose that we fit a model S = mn so that we expect effect of real-world networks deviating from the/ constant Ækæ 0:5 ai&mni . For the range of n encountered here – approximately assumption using the following iterative procedure: (i) regress n vs m four orders of magnitude – ai would therefore have to range over 2 for the data-set (as in Figure 2C); (ii) select network to remove from orders of magnitude. For this to occur, there must be sufficient data-set based on regression outcome; (iii) regress n vs S for reduced 2 variation in C and b , and these two quantities should correlate data-set and record new goodness-of-fit (as r ); (iv) repeat from (i) until i i 50% of networks removed. We do this for 3 selection cases, based on SD well with n. The ranges of the two variables are reasonably large in D # # # # here, in step (ii): (a) removing the network with the largest deviation the data-set – using C , 0.209 bi 2.52 and 0.005 Ci 0.72 – from m n linear model increased the goodness-of-fit (#) for the S n and could plausibly generate the required 100-fold variation. linear model/ (the dotted line indicates the original goodness-of-fit value/ 2 However, the correlation coefficients with n are very small: for bi, at r = 0.78); (b) removing networks with the smallest deviation from 2 2 m n linear model decreased the goodness-of-fit (,) for the S n linear r = 0.028 and for Ci, r = 0.025. This would therefore appear to / / preclude a nonlinear relationship between SD and n for the model; (c) random deletion did not consistently change the goodness- of-fit (N). Thus deviation from the assumption of constant Ækæ correlates networks studied here. with deviation from the linear S n model for the real-world networks, To study the effect of a lack of correlation between n and as predicted by the WS model. In/ addition, case (c) shows that the linear D network parameters like Ci on linear scaling between S and n,we S n model is robust to taking random sub-sets of the networks. / ws ran a Monte Carlo simulation (see Materials and Methods). Each Identical trends were obtained for S . All random deletion data-points one of 1000 experiments consisted of sampling 27 randomly averaged over 1000 realizations of the regress-delete-regress sequence; both largest and smallest deviation cases were unique sequences. drawn C values for networks with constant b, and with a spread of doi:10.1371/journal.pone.0002051.g003 n over 4 orders of magnitude. For each network its small-world- ness was computed and a linear regression of S against n performed. This resulted in a mean of r2 = 0.8960.06 s.d. across region. Detailed network data for these are not available because of the great technical difficulties in reliably reconstructing even the 1000 experiments, showing the strong tendency for linearity in small networks such as the 302 C. Elegans nervous system this case. [27]. Indeed, high edge density itself may be the primary cause of The linear relationship is, however, sensitive to deviations from technical problems in reconstructing complete systems from many the approximation that Ækæ is constant. That is, networks that domains, resulting in their absence from the network literature. deviate furthest from the linear m n model in turn deviate furthest Nonetheless, approximate reconstructions can be attempted. from the linear S n model (Figure/ 3). This was shown using a Quantitative anatomical models of individual regions suggest novel regress-delete-regress/ procedure outlined in Materials and that each of the hundreds of thousands or millions of Methods (we were able to directly test the sensitivity to Ækæ, rather receive many thousands of connections, and each themselves than using a Monte Carlo approach as above, because the strong connect to similar numbers of target neurons [16,28]. Such correlation of m with n provided a baseline from which we could networks of neurons can have very low small-world-ness values for Æ æ quantify deviation of k from constancy). Further, Figure 3 shows their size [16], and thus fall far from the linear S n model that if we delete a random set of networks from the data-set, then discovered here. / the average effect is to not change the fit to the linear S n model: / We conclude here that the linear relationship between small- the linear scaling is robust, and does not depend on a specific world-ness and system size does not hold for an arbitrary collection network set. of networks, but is highly likely if all such networks have a similar The sensitivity of small-world-ness linearity with n to degree Ækæ mean node degree. suggests that introduction of networks with very high edge density into our sample would destroy the linear scaling. We can rewrite Other scaling properties of small-world-ness (8) using Ækæ =2m/n Having established that S scales linearly with n, it is also instructive to look at how its ratios scale with n.We SkT^jn{1, 20 find, as expected, that most networks falling into the small-world ð Þ class have approximately the same mean shortest path length as and see that mean degree scales linearly with edge density. Thus, a their equivalent E–R random graphs, and so l<1. Given this, it is network with high edge density implies high mean degree, which unsurprising that both cD and cws then scale linearly with n (see in turn would fall far from the linear S = an model, as we have just Figure S1). We did find that three networks in our data-set — shown. email messages (#7), software packages (#21), and software classes One exemplar of a real system with high edge density is the (#22) — had l<0.1, indicating that their mean shortest path network of individual neurons within a single vertebrate brain length was an order of magnitude smaller than the equivalent E–R

PLoS ONE | www.plosone.org 6 April 2008 | Volume 3 | Issue 4 | e0002051 Network ‘Small-World-Ness’ random graph. These networks are thus ultra-small [29], and A noisy, limited growth process can generate the specific linear indeed both email message (#7) and software package (#21) S = an relationships we report. A generalized form of the Klemm- networks fall further from the linear model than any others. Eguiluz model (GKE)[34,35] encapsulates this process, and has Given the existence of the linear scaling with n, the scaling of the unique property of creating networks that are both small-world small-world-ness with some other topological properties is (short path length, high clustering) and ‘scale-free’ (having a completely determined. We can directly determine from (20) truncated power-law degree distribution) as found for many real- how edge density behaves in our data-set (values for j are given in world systems considered here. (To the best of our knowledge, all Table 1). Taking our fitted linear model S = an, we can substitute known real-world systems with power-law-like degree distributions n = S/a in (20) and find that also fall into the broad ‘small-world’ class we discussed in the Introduction; it is only the scale-free networks formed by the simple models that form a distinct set of ‘scale-free-only’ networks). S^SkTaj{1: 21 ð Þ By using the GKE model, we therefore also show that linear scaling of S can occur whether or not the real-world systems have Substituting our found values of Ækæ and a for the fits to either Sws ‘scale-free’ properties. or SD confirms that this is a good approximation. Therefore, The GKE model begins with an active set of M nodes. At every because small-world-ness linearly scales with network size, and time-step a new node is added, connecting d edges: one edge degree is approximately constant, then S also has a simple inverse added to a random inactive node with probability r, adding noise linear scaling with edge density. to the process; all remaining edges connect to randomly chosen active nodes. One of the active nodes is deactivated with a Real-world systems do not maximize small-world-ness probability proportional to the active nodes’ degrees; finally, the We can show that the specific scaling coefficient a in the new node is activated. The sequence repeats until the desired size relationship S = an for the real-world networks studied here does of network is obtained. not maximize small-world-ness for a particular size of network. We found that specific values for M and r could generate GKE First, we show that the WS model predicts an approximately networks with the same linear scaling relationships between D constant amount of rewiring p that maximizes S , independent of network size and Sws and SD that we observed for the real-world network size. To do this, given the above analytic expressions (13) networks (Figure 4; see Materials and Methods, and Figure S2). D & for lws and (16) for cws (and again assuming nKp 1), we found Therefore, a possible general mechanism for particular linear D D ~ dSws dp, and set dSws dp 0. Solving this equality for p would scaling rates of small-world-ness is a common size of both active D then give us the value of p that maximized Sws, if one existed — node set and quantity of noise during creation of the real-world see Text S1 for details of the solution. systems. We did this over the range n[ 103,1020 with K = Ækæ/2 = 2.5, * since this is implied by the result ofÂÃ Figure 2C. If p is the value of p Discussion D 3 giving maximal Sws, we found that for small networks (n =10 ) p* = 0.222 and, as nR‘, p*R0.246, so that the range of p* is very Small-world-ness is a topological property linking real-world small. The constant K and very small range of p* imply that the systems across domains of research. Hitherto it has been defined D associated maximum Sws values should scale linearly with n.It only in semi-quantitative way (Definition 1). In this paper we D SD Sws transpires that the theoretical maximum Sws depends almost propose quantitative measures of small-world-ness – and – exactly in a linear way on n with slope 0.181 (and plotted in and define a network to be in the small-world category with Figure 2A). Thus, SD is not maximized by the real-world networks. respect to either of them if the small-world-ness is greater than 1

A generative mechanism for a specific linear S n relationship / We have established and explained many simple properties of real-world networks and of their equivalence class in the WS model. We now show how the specific, sub-maximal, linear scaling of S = an could have been generated. The models we examine here are intended as informative examples of the generation and limits on S scaling, not an exhaustive list of those which could generate the specific linear scaling we found — that remains the subject of future work. Many of the real-world systems share common generative principals despite their widely differing origins. Most systems have a growth process, showing some form of [30] that is limited by the cost of adding new edges and by the Figure 4. A limited growth process can generate the observed S n relationships. capacity to maintain them (as might be induced by aging) [31]. We created 5 networks for every value of n[/500,750,1000,1500,2000 using a generalized form of the Klemm- Simple models of this process result in ‘scale-free’ networks with Eguiluz½Š model (GKE) [34,35], setting d = 3 so that the resulting networks power-law or truncated power-law degree distributions [30,31], a had the same mean degree (,6) as we found for the real-world property that is also common to many real-world systems systems. We searched on the GKE model’s M and r parameters (see considered here [32] (but see [33] for an alternative view of some text) to minimize the root mean square error between the resulting ws N D # biological networks). However, networks generated by these scaling of the averaged S ( ) and S ( ) values and the observed real- world network relationships (respectively, the solid and broken lines). models are not ‘small-world’ by either Definition 1 or 2. Their The best-fit parameters were different for the two forms of S, clustering coefficient is inversely proportional to n, going to zero as underlining that they measure two different properties of the real n grows large [34]. Thus, they cannot show linear scaling of S:itis systems. at best constant and at worst goes to zero with increasing n. doi:10.1371/journal.pone.0002051.g004

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(e.g Definition 2). This quantification of small-world-ness allows occurred for progressively smaller p as the oscillators’ symmetric for the statistical testing of its presence in any given network. coupling strength increased (for fixed n, K). Therefore, if a The Watts-Strogatz (WS) model plays a key role in the study of system’s function required it to be at the phase transition, so that small-world networks. It uses a generative process to create classes it could rapidly switch between synchronised and desynchronised of small-world network and is now widely used as a model for states with minimal perturbation, the required amount of studying dynamic systems [3–13]. However, until now, a precise (implied) rewiring may be far from that which maximised parameterization of the WS model associated with a given kind of small-world-ness. real-world network remained elusive. Our introduction of a These are just a few of many possible explanations for why real- quantitative measure of small-world-ness remedies this by world systems do not maximise small-world-ness. Instead we might demanding that the WS counterpart to a specific network have ask, when would small-world-ness be maximised? Maximum S the same value of SD (or Sws). For the WS models it is possible to essentially identifies the point in the network’s possible topologies show analytically that, under certain circumstances (constant re- where the highest clustering is achieved for the smallest deviation wiring parameter and range), the small-world-ness SD will scale from the shortest mean path length. Such a network would be linearly with network size n. Intriguingly, a wide class of real-world optimal for message-passing, such that all the nodes receive a networks also shows this linear scaling. Given this similarity in message in the shortest possible number of network steps [40]. On behavior, the assumption of (limited) topological correspondence this basis, we expect that some form of dynamic phenomenon, of the WS model with real networks implies certain constraints on whether based on percolation (or, equivalently, epidemiological empirically measured parameters (like mean degree) of these SIR models), oscillators, or some other general ordinary networks. These constraints appear to hold, and so the ideas differential equation system, will have a strong correlation with developed here provide further support for using the WS model in small-world-ness. So, just as we have used a continuously graded the study of small-world systems. ‘small-world-ness’ to quantitatively examine the topologies of the We have shown that the linear scaling between SD and n is not broad class of small-world networks, we may use this as the starting an inevitable property of networks; it would be possible, for point for quantifying the continuum of dynamic properties that example, to include networks with very large edge density that must also span this class. would destroy any linear scaling. However, in the event of linear scaling, there is a variety of possible scaling constants and there is a Materials and Methods noisy growth process that could give rise to the networks sharing the same scale (slope) parameter. Finally, we have shown that the Data-set of real-world systems small-world networks used here do not maximize SD (there are We collated a database of real-world networks’ topological ‘steeper’ linear relationships between SD and n). properties, combining published results with our own analyses of We cannot, on the basis of the work presented here, answer the available data-sets. These are presented in Table 1, extending the question of why small-world-ness was not maximised, but we can previous considerable effort of collating topological properties by give some insights as to why this is the case. The possible Mark Newman [1]. All networks are treated as undirected. We list ws explanations split into two broad classes of structural and 33 real-world systems in total: we could compute S for all systems D D dynamical limitations. Our use of the GKE model showed that and S for 27 systems — C could not be found or computed for the limited capacity of a system’s nodes to maintain edges (whether those systems. due to physical cost, aging processes, or some other mechanism) is We emphasise that the networks were not chosen for their ability one structural limitation that could result in sub-maximal small- to fit the linear model of S n. The majority of the data-set (21 of 33) / world-ness. Other structural limitations could include physical were obtained from a previous collation [1]: networks were only D limits on node location and length of edges, such as might occur omitted from that prior data-set if neither Cws or C were available for the sub-stations and transmission wires in the power grid for them (and hence were of no practical use to us here). Many of the network. additional networks we added filled sub-domains missing from the Even if structural limitations were not an issue, then the system prior data-set, for example: the dolphin network [41] is an example may have dynamical requirements that prevent it from maximis- of an animal ; the cortical area connectivity map [42] ing small-world-ness. The constraints placed on a system’s is an example of large-scale neural connectivity. In addition, the topology by the dynamics required to fulfill its function are not regress-delete-regress sequence we used in the main text (and see well understood. Recent work has shown how the presence of below) shows two properties. First, that we could have applied that particular network ‘motifs’ — repeating patterns of connections method to the data-set in Table 1 before further analysis, pruning the between a small number of nodes — can guarantee, for example, a data-set down to those networks that showed the best fit to the linear chaotic attractor for the network as a whole [36]. The functional model (by choosing the ‘most-deviant’ networks to omit), but did not. requirements of some real-world system may then lead to the Second, that the linear scaling property is robust across randomly inclusion of particular motifs to guarantee the necessary dynamics chosen sub-sets of the network data-set: on average, randomly [37], and there is no necessary link between a system’s motifs and deleting networks from the data-set did not significantly reduce the fit its global topological properties (of which small-world-ness is but to the linear model. one). Nonetheless, given that so many of the key motifs identified so far are either complete 3-node loops or contain them [37,38], Testing significance of S scores the global topology will have a high clustering coefficient, and will We assess the significance of borderline small-world-ness most likely be a small-world network. scores S1 using Monte Carlo methods. The null hypothesis for Other systems may have constraints placed directly on their the Watts-Strogatz definition of small-world networks is that the global topology, and this too could prevent maximisation of system is an Erdo¨s-Re´nyi (E–R) random graph. We thus small-world-ness. For example, in his original work on the small- constructed N = 1000 E–R networks with the same number of world model, Watts [39] explored the dynamics of Kuramoto nodes n and edges m for each tested real-world system, D ws oscillators on a WS model substrate, and showed that the computing Si and Si for the ith E–R network. The 99% fraction of synchronised oscillators had a phase transition that confidence limits for the null hypothesis were then defined for

PLoS ONE | www.plosone.org 8 April 2008 | Volume 3 | Issue 4 | e0002051 Network ‘Small-World-Ness’ each system. We first found the central 99% interval [a*, b*], Monte Carlo testing of linear scaling that is [43] We tested the dominance of linear S n scaling given an approximately constant mean node degree/Ækæ and path length D D scaling b. For each Monte Carlo simulation, we drew 27 random # S vaà # S vbà i ~0:005, i ~0:995, 22 C values from a uniform distribution in [0,1], computing a for ÈÉN ÈÉN ð Þ i each from Eq. (19) with constant bi = 1 and Ækæi = 6. We then ws and similarly for S . The 99% confidence interval for the system computed Si = aini for each, using a logarithmic spread of 27 is then network sizes n[ 102,106 to closely match the spread of the real- world system sizes.Âà Linear regression (as detailed above, including the log -transform) was then performed on the set of simulated 27 b {a 10 CI~ à à : 23 S values and the r2 recorded. We repeated this procedure 1000 2 ð Þ times. The upper 99% confidence limit is then CL0.01 =1+ CI (where by definition Sws, SD = 1 for an E–R random graph). A network Searching GKE model parameter space with S.CL0.01 was therefore considered to significantly differ We wished to determine if the generalized Klemm-Eguiluz from a random network. We note that adopting a quantitative model (GKE)[31,32] model could explain the particular scaling definition (Definition 2) of small-world-ness has led us to a relationships we found for the real-world systems: procedure for a general statistical test for the presence of small- world structure, as defined by Watts and Strogatz [3], which is SD~0:023n0:96, 24 particularly useful for establishing meaningful departures from ð Þ randomness in small networks. Sws~0:012n1:11: 25 Fits to linear scaling ð Þ Least-squares regressions on small-world-ness S and number of edges m against size of system n were performed on log10- We explored the (M, r) parameter space, searching over transformed data to normalize magnitude of errors across range M[ 3,30 in steps of 1, and r[ 0,0:5 in steps of 0.1. The lower ½Š ½Š of n.Bestfitlinearmodellog10(x)=a+blog10(n) back-transformed limit on M is set by the number of edges added per new vertex, to a linear basis, giving x = anb,wherea =10a and b = b. and here we set d = 3 to give a mean degree of Ækæ<6 for a GKE MATLAB (Mathworks) function regress was used to perform model network, approximately the same degree that was implied the regressions. The validity of r2 significance values was by the linear m n relationship for the real-world systems. For each / established by confirming that the residuals of each regression (M, r) pair, we constructed 5 GKE model networks for each value of n[ 500,750,1000,1500,2000 and computed their Sws and SD had a normal distribution at p = 0.01 using the Anderson-Darling ½Š test [44]. scores. We took the mean of these 5 scores for each n, giving sets ws ws ws ws ws D D D D D SS500,SS750,SS1000,SS1500,SS2000 and SS500,SS750,SS1000,SS1500,SS2000 . A regress-delete-regress procedure for testing TheÈÉ fit of the GKE model networksÈÉ was then assessed by robustness of predictions computing the root mean square error (RMSE) between these We test the effect of real-world networks deviating from the mean values and those given by the scaling relationships (24) and constant Ækæ assumption using the following iterative procedure: (25) for the tested sizes of network n. The parameters that minimised RMSE are given in Figure 4; the error landscapes are 1. regress n vs m for the data-set (as in Figure 2B); shown in Figure S2. 2. select network to remove from data-set based on regression outcome (3 different selection criteria were used, detailed Supporting Information below); Text S1 Supporting information text 3. regress n vs S for reduced data-set and record new goodness-of- 2 Found at: doi:10.1371/journal.pone.0002051.s001 (0.22 MB fit (as r ); PDF) 4. repeat from step 1 until 50% of networks removed. Figure S1 Correlation of real-world systems’ clustering coeffi- We do this for 3 selection cases in step 2. First, we tested cient and path length ratios with system size. Clustering coefficient D D ws ws removing the network with the largest deviation from m n linear ratios (a) c = C Crand and (b) c = C Crand both scaled linearly 2 model in each iteration, hypothesizing that this should lead/ to an with S. Linear regressions found r >0.86 in both cases. (c) As overall increase in fit to a linear model (increased r2) for S = f(n)if expected for small-world-networks, path length L was approxi- the WS model behaviour reflected that of real-world systems. mately the same as that of an E-R random graph, and so l = L/ Second, we tested removing the network with the smallest Lrand1 for most networks (note that we show l here for all 33 deviation from m n linear model at each iteration, hypothesizing networks). All linear regressions performed on log10-transformed that this should lead/ to an overall decrease in fit to a linear model data, as detailed in Materials and Methods of the main text. (decreased r2) for S = f(n) if the WS model behaviour reflected that Found at: doi:10.1371/journal.pone.0002051.s002 (0.11 MB TIF) of real-world systems. Third, we tested random deletion, where a Figure S2 The root mean square error (RMSE) distribution random network was deleted at each iteration, irrespective of the across tested values of the GKE model parameters. The RMSE is regression outcome, to establish the baseline effect of removing computed based on the difference between the mean values of systems from the data-set. The first and second cases are unique small-world-ness for a set of generated GKE networks and the sequences of deleted networks; the third case we repeated 1000 corresponding small-world-ness values from the specific linear times. relationships found for the real-world systems. (a) RMSE error

PLoS ONE | www.plosone.org 9 April 2008 | Volume 3 | Issue 4 | e0002051 Network ‘Small-World-Ness’ distribution for the fit to the Sws n relationship. RMSE plotted on Acknowledgments log scale to emphasise valley of/ minimum values. Stick-and-ball indicates the parameter pair that minimised RMSE. (b) RMSE We thank M. Newman and M. Kaiser for making their data-sets publicly D available, and T. Stafford and T. Prescott for comments on drafts of this error distribution for the fit to the S n relationship. Stick-and- manuscript. ball indicates the parameter pair that minimised/ RMSE. Found at: doi:10.1371/journal.pone.0002051.s003 (0.44 MB TIF) Author Contributions Conceived and designed the experiments: MH KG. Performed the experiments: MH. Analyzed the data: MH KG. Contributed reagents/ materials/analysis tools: MH. Wrote the paper: MH KG.

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